Evaluate a polynomial when coefficients are multi-dimensional in Python

To evaluate a polynomial at points x with multi-dimensional coefficients, use the numpy.polynomial.polynomial.polyval() method in Python. This method handles coefficient arrays where multiple polynomials can be stored in different columns.

Parameters

The polyval() method accepts three key parameters ?

• x ? The points at which to evaluate the polynomial. Can be a scalar, list, or array
• c ? Array of coefficients where c[n] contains coefficients for degree n terms. For multidimensional arrays, columns represent different polynomials
• tensor ? If True (default), evaluates every column of coefficients for every element of x. If False, broadcasts x over columns

Creating Multi-dimensional Coefficients

First, let's create a multi-dimensional coefficient array and examine its properties ?

import numpy as np
from numpy.polynomial.polynomial import polyval

# Create a multidimensional array of coefficients
c = np.arange(4).reshape(2,2)

# Display the array
print("Our Array...\n", c)

# Check the Dimensions
print("\nDimensions of our Array...\n", c.ndim)

# Get the Datatype
print("\nDatatype of our Array object...\n", c.dtype)

# Get the Shape
print("\nShape of our Array object...\n", c.shape)

Our Array...
 [[0 1]
 [2 3]]

Dimensions of our Array...
 2

Datatype of our Array object...
 int64

Shape of our Array object...
 (2, 2)

Evaluating the Polynomial

Now we'll evaluate the polynomial at points [1, 2] using the multi-dimensional coefficients ?

import numpy as np
from numpy.polynomial.polynomial import polyval

# Create coefficient array: [[0,1], [2,3]]
# This represents two polynomials:
# Column 0: 0 + 2x (coefficients [0,2])
# Column 1: 1 + 3x (coefficients [1,3])
c = np.arange(4).reshape(2,2)

# Evaluate at points x=[1,2] with tensor=True
result = polyval([1,2], c, tensor=True)
print("Result with tensor=True:\n", result)

# Each column of c is evaluated for each point in x
print("\nExplanation:")
print("For x=1: Column 0: 0+2(1)=2, Column 1: 1+3(1)=4")
print("For x=2: Column 0: 0+2(2)=4, Column 1: 1+3(2)=7")

Result with tensor=True:
 [[2. 4.]
 [4. 7.]]

Explanation:
For x=1: Column 0: 0+2(1)=2, Column 1: 1+3(1)=4
For x=2: Column 0: 0+2(2)=4, Column 1: 1+3(2)=7

Understanding the Output

The result matrix has shape (2, 2) where ?

• Rows correspond to evaluation points [1, 2]
• Columns correspond to the different polynomials
• Each element result[i,j] is polynomial j evaluated at point x[i]

Comparison: tensor=True vs tensor=False

import numpy as np
from numpy.polynomial.polynomial import polyval

c = np.arange(4).reshape(2,2)

# With tensor=True (default)
result_tensor = polyval([1,2], c, tensor=True)
print("tensor=True shape:", result_tensor.shape)
print("tensor=True result:\n", result_tensor)

# With tensor=False
result_broadcast = polyval([1,2], c, tensor=False)
print("\ntensor=False shape:", result_broadcast.shape)
print("tensor=False result:", result_broadcast)

tensor=True shape: (2, 2)
tensor=True result:
 [[2. 4.]
 [4. 7.]]

tensor=False shape: (2,)
tensor=False result: [2. 7.]

Conclusion

Use polyval() with multi-dimensional coefficients to evaluate multiple polynomials simultaneously. The tensor parameter controls whether to evaluate all polynomials at all points (True) or broadcast for element-wise evaluation (False).